A117683 Triangle T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), read by rows.

1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 1, 4, 4, 1, 1, 6, 6, 24, 6, 6, 1, 1, 6, 6, 6, 6, 1, 1, 8, 8, 48, 12, 48, 8, 8, 1, 9, 72, 72, 108, 108, 72, 72, 9, 1, 10, 90, 720, 180, 1080, 180, 720, 90, 10, 1, 1, 10, 90, 180, 180, 180, 180, 90, 10, 1, 1, 12, 12, 120, 270, 2160, 360, 2160, 270, 120, 12, 12, 1

Offset: 1

Author

Roger L. Bagula, Apr 12 2006

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,  1;
  4,  4,  4,   1;
  1,  4,  4,   1,   1;
  6,  6, 24,   6,   6,  1;
  1,  6,  6,   6,   6,  1,  1;
  8,  8, 48,  12,  48,  8,  8,  1;
  9, 72, 72, 108, 108, 72, 72,  9,  1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A049614, A117684.

Programs

Magma

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
A117683:= func< n,k | A049614(n)/(A049614(k)*A049614(n-k)) >;
[A117683(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 21 2023

Mathematica

f[n_]:= If[PrimeQ[n], 1, n];
cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A049614 *)
T[n_, k_]:= T[n, k]= cf[n]/(cf[k]*cf[n-k]);
Table[T[n, k], {n,12}, {k,n}]//Flatten

PARI

primorial(n)=prod(i=1,primepi(n),prime(i))
T(n,m)=binomial(n,m)*primorial(m)*primorial(n-m)/primorial(n) \\ Charles R Greathouse IV, Jan 16 2012

SageMath

def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
def A117683(n,k): return A049614(n)/(A049614(k)*A049614(n-k))
flatten([[A117683(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 21 2023

Formula

T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), for 1 <= k <= n, n >= 1.

Sum_{k=1..n} T(n, k) = A117684(n).

Extensions

Edited by the Associate Editors of the OEIS, Aug 18 2009

Edited by G. C. Greubel, Jul 21 2023